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IF a 1 , a 2 , a 3 ….….…...a n , A 1 , A 2 , A 3 ….…..,A n , K are all real numbers , then prove that (A 1 2 ÷ (x-a 1 ) ) + (A 2 2 ÷ (x-a 2 ) ) +......................... (A n 2 ÷ (x-a n )) = K has no imaginary roots .

IF a1 , a2 , a3 ….….…...an , A1, A2 , A3 ….…..,An , K are all real numbers , then prove that
(A1 2 ÷ (x-a1) ) + (A2 2 ÷ (x-a2) ) +......................... (An 2 ÷ (x-an)) = K has no imaginary roots .

Grade:12th pass

1 Answers

mycroft holmes
272 Points
9 years ago
Suppose z is a non-real root of the given equation i.e. 
 
 
Then by taking conjugates we see that 
 
 
Subtracting the two equations we get
 
 
Since, z is non-real we have    and hence
 
 
which is not possible unless all Ai are zero.
 
Hence the given equation has no non-real root.
 
 

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